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Partition of Unity Finite Element Analysis of Nonlinear Transient Diffusion Problems Using p-Version Refinement

Abdelkarim El Kahoui1, Mustapha Malek1, Nouh Izem1, M. Shadi Mohamed2, Mohammed Seaid3,4,*
1 Laboratory of Engineering Sciences, Faculty of Science, Ibn Zohr University, Agadir, Morocco
2 School of Energy, Geoscience, Infrastructure and Society, Heriot-Watt University, Edinburgh, EH14 4AS, UK
3 Department of Engineering, Durham University, Durham, DH1 3LE, UK
4 International Water Research Institute, University Mohammed VI Polytechnic, Benguerir, Morocco
* Corresponding Author: Mohammed Seaid. Email:

Computer Modeling in Engineering & Sciences 2020, 124(1), 61-78.

Received 03 April 2020; Accepted 29 April 2020; Issue published 19 June 2020


We propose a high-order enriched partition of unity finite element method for linear and nonlinear time-dependent diffusion problems. The solution of this class of problems often exhibits non-smooth features such as steep gradients and boundary layers which can be very challenging to recover using the conventional low-order finite element methods. A class of steady-state exponential functions has been widely used for enrichment and its performance to numerically solve these challenges has been demonstrated. However, these enrichment functions have been used only in context of the standard h-version refinement or the so-called q-version refinement. In this paper we demonstrate that the p-version refinement can also be a very attractive option in terms of the efficiency and the accuracy in the enriched partition of unity finite element method. First, the transient diffusion problem is integrated in time using a semi-implicit scheme and the semi-discrete problem is then integrated in space using the p-version enriched finite elements. Numerical results are presented for three test examples of timedependent diffusion problems in both homogeneous and heterogeneous media. The computed results show the significant improvement when using the p-version refined enriched approximations in the finite element analysis. In addition, these results support our expectations for a robust and high-order accurate enriched partition of unity finite element method.


Partition of unity; finite element method; p-version refinement; enrichment functions; nonlinear diffusion problems

Cite This Article

Kahoui, A. E., Malek, M., Izem, N., Mohamed, M. S., Seaid, M. (2020). Partition of Unity Finite Element Analysis of Nonlinear Transient Diffusion Problems Using p-Version Refinement. CMES-Computer Modeling in Engineering & Sciences, 124(1), 61–78.


This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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